Why did the Doctor think he was going to die in "The Time of the Doctor" even though he saw Capaldi in "The Day of the Doctor"?

TL; DR: Yes, it's possible that during Time of the Doctor, the Eleventh Doctor has forgotten his future self's involvement in the saving of Gallifrey.

To add to the answer by Hypnosifl and to answer the specific part of your question "Could he have just forgotten?", remember that the War Doctor and the Tenth Doctor (and by extension, the Ninth) don't retain the memories because of the muddled time streams, but the Eleventh does, because it's his present. Capaldi's appearance, though, would be from his future, and so it's entirely likely that he doesn't remember it for the same reason.

Maybe it's that Time Lords can't generally retain memories of interactions with their future selves. This is a consequence of the Blinovitch Limitation Effect shown with the Brigadier in Mawdryn Undead -- the younger version can't remember the interaction with the older. It also explains why the Fifth Doctor doesn't recall being pulled into the Death Zone in The Five Doctors despite having been there three times previously. (I say "generally" because it seems like the Second Doctor remembers the events of The Three Doctors during his conversation with the Brigadier in The Five Doctors. However, since the Time Lords deliberately brought the three incarnations together, they may have had some way to circumvent the effect.)

Granted, the events of Time Crash seem to contradict that since it implies that the Fifth Doctor would remember watching the Tenth fix the TARDIS, but that could be explained by the memories of the younger simply being suppressed until he "catches up" with the future events. So he does remember, he just doesn't remember that he remembers until he's on the other side of the conversation; that is, when he's Ten.

This also accounts for Eleven's line in Day of the Doctor, "This is where I come in." That implies that he remembers meeting his future self when he was Ten, in spite of explicitly saying that Ten won't remember. The memories are still there, just inaccessible until the most forward point in the timeline. At the point where the time streams cross, Eleven suddenly has access to the memory his past self made, leading him to remark, essentially, "Oh, right! This happened. I'd forgotten."